Extended TFTs

So I’ve finally managed to bang my dissertation into something more or less ready for public consumption. It is basically finished (except for some typos and spell checking).

It is available on my new website.

The title is “The Classification of Two-Dimensional Extended Topological Field Theories”.

I will continue to provide updates and improvements as they happen.


We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these types of 2-dimensional extended topological field theories with arbitrary target bicategory. As an immediate corollary we obtain a concrete classification when the target is the symmetric monoidal bicategory of  algebras, bimodules, and intertwiners over a fixed commutative ground ring. In the oriented case, such an extended topological field theory is equivalent to specifying a separable symmetric Frobenius algebra.

Along the way we collect together the notion of symmetric monoidal bicategory and define a precise notion of generators and relations for  symmetric monoidal bicategories. Given generators and relations, we prove an abstract existence theorem for a symmetric monoidal bicategory which satisfies a universal property with respect to this data. We also prove a theorem which provides a simple list of criteria for determining when a morphism of symmetric monoidal bicategories is an equivalence. We introduce the symmetric monoidal bicategory of bordisms with structure, where the allowed structures are essentially any structures which have a suitable sheaf or stack gluing property.

We modify the techniques used in the proof of Cerf theory and the classification of small codimension singularities to obtain a bicategorical decomposition theorem for surfaces. Moreover these techniques produce a finite list of local relations which are sufficient to pass between any two decompositions. We deliberately avoid the use of the classification of surfaces, and consequently our techniques are readily adaptable to  higher dimensions. Although constructed for the unoriented case,  our decomposition theorem is engineered to generalize to the case of bordisms with structure. We demonstrate this for the  case of bordisms with orientations, which leads to a similar classification theorem.

10 thoughts on “Extended TFTs

  1. It is probably just my own opinion, but I think if you had looked at our AMS book, or the paper by Carter Rieger and Saito on movie moves, you might have saved yourself some trouble in the first chapter.

    A lot of the singularity theory that you use is kind-of-sort-of known to singularity theorists. It is really hard to get a handle on their language, it is better to talk to them in person.

    For those who are trying to make progress in higher dimensions, there is a paper by David Mond and a student of his — I am pretty sure it is on the arxiv, but my homeweb is slow tonight — that classifies complex multigerms. There are always real pictures of these — complex quadratics give hyperbolic and parabolic confluences simultaneously. Anyway, an appropriate and detailed reading of Mond gives the Reidemeister moves in higher dimensions. These are usually an essential ingredient to the higher categorification.

    Morally, a Reidemeister move in dimension n is a measurable in dimension n+1. One has to keep track of the interaction of critical points and crossings as well.

    You will see the same moves on the sphere eversion links on the page listed below.

  2. It looks like an interesting thesis! I’ve been wanting to see a proof of your main theorem for over a decade. Of course a large part of the problem is stating the theorem precisely… but then it still takes a lot of singularity theory to prove it.

    I’m also very happy you worked out all these facts about symmetric monoidal bicategories and presented all the necessary definitions in a clear and well-organized fashion. This is math lots of people will need in the coming century – not just people working on TQFTs.

    So, congratulations!

    A couple typos: On page 9 you speak of Chern-“Simmons” theory. In the subject header for section 4.6 you mention “extendeded” TFTs.

  3. Hi Chris, congratulations again.
    I think one of the diagrams in the equality (SMA) saying that a sylleptic monoidal bicategory is symmetric is not correct. At least I don’t understand the non-labeled 2-morphism on the left hand side.

  4. Looking at it again, I think I found the right correction:
    on both sides of the equation, the right hand vertical arrow has to go upwards.

  5. Martin,

    You are exactly right. Thanks for catching that mathematical error/typo. I guess I got a little over zealous with my commutative square hot keys. Hopefully this is the only diagram error, but given the number of diagrams and figures, there could be more.

    Thanks everyone for sending me the typos you’ve found. I’ve fixed them in a new version, which I’ll post as soon as the google server comes back online for me.

    Due to graduation and moving to Germany, I’m going to be taking a two week internet hiatus. After I’m settled in Bonn, I’ll try to get out a non-double spaced version.

  6. Congratulations!

    * on the first line of Appendix A, you wrote “is fairly standard are standard”

    * in item 6 of Proposition A.2.4, you say that a module is called reflexive if it’s isomorphic to its dual, but I think it is “bidual” instead

  7. In the introduction you mention that one reason to be interested in extended topological field theories is that they are inherently more computable (you can chop the manifold into more elementary pieces).

    It struck me for the first time that this is precisely what Witten did when he computed the 3-manifold invariants in Chern-Simons theory nonperturbatively in his original paper “Quantum field theory and the Jones polynomial”.

    Namely, he argued heuristically that the Hilbert spaces of Chern-Simons theory in 3 dimensions are the same as the spaces of conformal blocks in the 2-dimensional WZW model.

    Now we know that the WZW model has to do with representations of loop groups, and that these are essentially the things which make up the category assigned to the circle. Some details though remain to be filled out as far as I am aware. At least I hope so because I hope to help in filling them out!

    So: Witten was using the _extended_ nature of Chern-Simons theory to work out the 3-manifold invariants! He was the first extended topological quantum field theorist, though he never knew it. Would you agree or am I subtly in error here? I think this is great, because it gives hope to the idea that understanding Chern-Simons theory as an _extended_ TQFT will help to actually calculate the 3-manifold invariants. This hopefully addresses the misgivings of Freed in “Remarks on Chern-Simons theory”. Moreover it may help to motivate extended TQFT to physicists: apparantly “extendedness” helps you to calculate things nonperturbatively.

  8. Was a bit surprised that you didn’t mention Nick Gurski’s thesis “An algebraic theory of tricategories”, since it seems relevant to (a) freely generating bicategories from certain initial data, and (b) monoidal bicategories…. even if you just wanted to say “the approach of Gurski will not work here, for such and such a reason”. As you are aware, Gurski gave the first totally algebraic definition of a fully-weak tricategory, in the sense that the original definition of Gordon, Power and Street contained some data which required choices (thus “nonalgebraic”) whereas Gurski’s definition included all the required data explictly spelt out.

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